When George and Fred are talking to Harry in the Dursley's house, the twins go from being in the back of the room to the front. Although these characters have been shown to be fond of Apparating short distances, at that point in the film, the house had had an anti-Apparation ward placed over it to prevent Harry from escaping the Death Eaters unnoticed.
In the Chamber of Secrets Dobby tells Harry that the rag he is wearing is the mark of the house elves enslavement. At the end of the movie Dobby is a free elf yet he is still wearing the rag in both Deathly Hollow movies. Dobby should be wearing fancy clothes as he does in the books.
(at around 2h) When Hermione stuns Harry to disguise him from the Snatchers, she takes his glasses off him and puts them in her own pocket. When Harry and Ron are put in the cellar at Malfoy Manor, Harry takes his glasses out of his pocket.
(at around 1h 35 mins) When Harry is preparing to jump into the lake after the sword of Gryffindor, he submerges, wearing his glasses. They are present as he drowns. After returning to the surface, all is blurry and he is no longer wearing them. Harry picks them up off the ice, from a position nowhere near the hole he dove through.
(at around 1h 23 mins) When Hermione and Harry are sitting on the rocks on the cliff the camera behind Harry shows Hermione putting down a book closed, when the cameras switch to behind Hermione it appears that the wind has blown the book open but the cameras switch back to show the book closed again, and switch one final time to see the book yet again open, flapping in the wind.
(at around 1h 50 mins) The piece of paper on which Xenophilius Lovegood draws the Hallows symbol changes between shots. When he is drawing the symbols there is a smudge on the paper, but when the shot pans away and he is talking to Harry, there is no smudge on the paper.
(at around 5 mins) In the first scene at Malfoy Manor, Voldemort asks Pius Thicknesse for his opinion on the current situation. In the first shot of Pius, the ministry-pin on his tie is seen with the M vertical. In the shot of him replying, the bottom of the M is tilted slightly to the right seen from camera POV. In the shot of him reacting to Voldemort saying he will prove useful, the M is vertical again.
(at around 48 mins) When Hermione casts Expelliarmus at Mundungus Fletcher and takes his wand (in Grimmauld Place), she and Harry seem to move straight forward in no time which clearly isn't Apparation.
(at around 20 mins) In the third book, Prisoner of Azkaban, Harry visits Lupin in his office where he sees a Grindylow. This does not happen in the third film (Harry Potter and the Prisoner of Azkaban). This creates a continuity error as Harry could not possibly know the answer to Lupin's question (What creature sat in the corner the first time Harry Potter visited my office in Hogwarts?) after they return to the Burrow following the Battle of the Seven Potters.
(at around 14 mins) After Fleur transforms into a replica of Harry she (as Harry) is seen standing with her shirt off wearing a bra. The bra's cup size is far too small and positioned far too high on her chest to have been worn while Fleur was still herself.
The Dursley house interior changes. When Mad Eye and the group show up to move Harry, there is burgundy wallpaper, carpet and dark cabinets. Later when the DeathEater shows up looking for Harry, everything is white and there is a wooden floor.
In the book, when Snape brings the sword, he was only able to find Harry and get past the security measures because Hermione took the portrait of Headmaster Phineas Nigellus Black from Sirius house and Black told Snape how to locate them while occupying his official Hogwarts portrait. In the movie they never took the portrait or interacted with it at all.
(at around 1h 7 mins) Just after Harry and Hermione fail to destroy the Horcrux and Harry says "We need to keep it safe", he looks inside the piece of his mirror, the camera is briefly seen in the reflection.
(at around 17 mins) When Hagrid and Harry emerge from the tunnel during the chase scene, they're on the North side of the Dartford crossing. The Queen Elizabeth II Bridge is on the left, and traffic is seen going into the tunnel. This would mean the traffic is heading South. This is unusual because the bridge usually carries the South-bound traffic, with the North-bound traffic split between the 2 tunnels. It's not unusual for one of the tunnels to be closed at night, so it's possible the production took advantage of this and created traffic where they could put a camera crew in a position that would obstruct the normal flow.
(at around 1h 28 mins) In Harry's flashback of the night his parents were killed, Voldemort is shown in his resurrected form opposed to his pre-death state of being. Neither the book nor the movie tell us exactly what Voldemort looked like the night of the Potters' murders. Both, however, allude to his changing appearance becoming less human.
(at around 1 min) When Hermione first appears in the movie she has a newspaper. The picture of the Muggle family does not move even though it is in a Wizard newspaper. However, one must assume that because it is, in fact, a Muggle family that they would not have had a Wizard picture taken of themselves, and the picture is a Muggle picture, hence the subject's stationary positions. It can be seen that the headline picture, however, does move.
During the flashback in which the Elder Wand is stolen from Gregorovitch, Grindelwald is never shown defeating him in order to gain the wand's allegiance. Since he never became its master, this means that Dumbledore could not have won it from him after their duel, thereby preventing Draco and eventually Harry from ever gaining mastery of it.
There was no need for the teachers and Harry's friends to go to Privet Drive and affect a rescue because they had the ability to 'disapparate' and appear elsewhere as Mundungus did during the fight on the brooms.
When Harry walks across the frozen pond, both in his shoes and when he is barefoot, there are no footprints left on the ice. The pond is covered with a light snow, that Harry brushes away to see the Sword of Gryffindor beneath the frozen surface. The snow would show any tracks.
(at around 1h 30 mins) When Hermione is reading Rita Skeeter's book, she tells Harry that she has discovered that the boy in the photo at Bathilda Bagshot's house is Gellert Grindelwald, she hands Harry the book. If you look closely, you can see there is a oval shaped hole in the front cover where the moving picture of Dumbledore would be.
The goof item below may give away important plot points.
Incorrectly regarded as goofs
(at around 1h 40 mins) After Ron destroys the necklace horcrux, he states that there are three more horcruxes left to find and destroy. At this point they thought there were only six horcruxes plus Voldemort, not seven.